This is different in different countries. Ukrainians generally start counting days of the week from Monday.
шостий is the adjectival form of the number six which are called ordinal numbers. Because they describe nouns, they match in gender (m/f/n), number (sing/pl), and case (nom/gen/etc). In this case, "шостий" is describing день (not Saturday) which is why it's in the masculine singular nominative.